VIII.
Prof. K. U. Ingard
Dr. L. W. Dean III
Dr. G. C. Maling, Jr.
Dr. H. L. Willke, Jr.
A.
PHYSICAL ACOUSTICS
A. V. Dralle
S. R. Ellms
S. T. Emerson
T.
F.
W.
J.
J.
A.
M.
M.
Kinzer III
Lidgeois
Manheimer
Witting
STABILITY OF PARALLEL FLOWS
Heisenberg's discovery of unstable asymptotic solutions for the separated perturbation equations for plane viscid Poiseuille flow, 1 together with Rayleigh's proof of the
nonexistence of unstable separating solutions in the inviscid case, has encouraged the
conjecture that viscous forces can be a cause of instability. When the rarity of separating solutions among the solutions of the full initial value problem for the perturbed
flow is properly taken into account, however, this apparent disconformity with the
expected stabilizing role of viscosity disappears. The absence of separating inviscid
solutions corresponding to the viscous ones found by Heisenberg is to be viewed as
implying not that the inviscid flow is stable, but only that the equations of perturbation
fail to separate in the inviscid case.
The equations of perturbation do seperate in general under perturbations that are
homogeneous in the direction of the basic flow; otherwise, in the case of inviscid flows
3
without points of inflection, they do not. In the author's thesis, the initial value problem restricted to such perturbations was worked out for an arbitrary basic flow in the
inviscid case, and for plane and cylindrical flows in the viscid case. A somewhat less
transparent approach was used earlier to discuss the general inviscid and plane viscid
cases.4 It is shown that every inhomogeneous inviscid flow is unstable under such perturbations. In the viscid case, the solutions follow the corresponding inviscid ones in
time until growth by a factor of the order of the Reynolds number is achieved, after
which rapid viscosity-controlled decay sets in. Comparison with the exact equations
of motion indicates that qualitatively different results are not to be expected when per3 4
turbations of arbitrary amplitude are admitted. '
In the author's thesis, 5 it is also shown in the plane Couette case, and proposed on
physical grounds in the general case, that the inviscid flow is actually not stable under
any perturbation at all. Asymptotic separating solutions are found for the viscid Couette
case; and these, in contrast with their inviscid counterparts, are all stable. In the
absence of boundaries, viscid Couette flow is shown, moreover, to be stable under every
sufficiently smooth perturbation.
5
A brief outline of the work on plane Couette flow is given here.
The fundamental perturbation equation for a plane parallel flow W(x l ) in the x 3
4
direction has been given in the form
*This work was supported in part by the U. S. Navy (Office of Naval Research) under
Contract Nonr-1841(42).
QPR No. 70
(VIII.
PHYSICAL ACOUSTICS)
a V2 u
at
1
= vV 4 u
1
-W
u
ax 3
1
+ d 2 W au
dx 2 ax 3
1
For the plane Couette flow W(x 1 ) = x 1 V, Eq. 1 simplifies to
- vV
2
+ Vx
V2 u 1
1
0.
(2)
The exact free-field propagator for (2) has been found to be
K(x,y,t) =
(4Trvt)3/2 (+
32
V2t2)1/2
22
exvt[(xy1)
S2
Sexp
4
xl-Y1)2
(x2 -y ) +
+ (x2+-Y22
x33 ++ Vt x
2222
1 + 12
Y3)
Y3
(3)
3
so that, formally, for t > 0,
V 2 ul(x,t) =
K(x,y,t) V 2 ul(y,0) d 3y.
(4)
The propagator (3) is similar to the familiar Poisson propagator for the heat equation, the result of the extra terms in (3) being ultimately, if anything, an increase in
the damping of the initial disturbance. Thus it is shown readily that in the absence of
boundaries the plane viscid Couette flow is stable under every twice -differentiable
perturbation ul(X,O) that tends to zero, along with its second derivative, for x2 -0.
When periodic boundary conditions are imposed in the x 2 and x 3 directions, the same
result holds for perturbations that are periodic rather than those that tend to zero in
those directions.
In the presence of boundaries, we require, not the free-field propagator, but a propagator maintaining viscous boundary conditions. The term Vt(x 1+yl)/2 in the exponential in (3) rules out the use of the image methods usually employed in constructing
bounded-field propagators from free-field ones.
In the absence of a conclusive demonstration of the stability of plane viscid Couette
flow in the presence of boundaries, we have sought unstable separating solutions for (2)
of the type
VZul(X, t ) = a(x) est exp(ik 2 x2 +ik 3 x 3 )
QPR No. 70
(5)
(VIII.
PHYSICAL ACOUSTICS)
for which (2) reduces to
s + vk 2 - v d
2
2
where k
+ ik 3 Vx)
(6)
a(x 1 ) = 0,
2
= k2 + k3
Equation 6 is a form of Airy's equation, the two basic solutions of which can be written as contour integrals by means of the Laplace method.
Accurate asymptotic repre-
sentations have been found for these integrals through the use of the saddle-point method
of integration.
The requirement that ul vanish, along with its first derivative,
boundaries, say xl = 0 and xl = 1, establishes an eigenvalue problem for s.
at given
The prob-
lem has solutions, the number and accuracy of which increase without limit as k 3 V/v =
k3R -
All such solutions for ul(x,t) are stable, and, in fact, decay with time at
oo.
least as rapidly as
exp [
vk
4
+
(7)
.
6
When v = 0, the basic equations of perturbation contain no derivatives of the u1
2
higher than the first, so that V ul need not exist.
V u l , does exist, however, they will satisfy the
perturbation u0), as well as a
3
inviscid form of (2):
+ Vx
When the Laplacian of the initial
V 2 ul =0.
a
(8)
2
Under the initial conditions V u(0) = a(xl,x2,x3), the relevant solution of (8) will
have the familiar form
V 2 ul(x 1 ,xZ,x
3
,t) = a(x 1 ,xZ,x
(9)
3 -Vx 1 t),
which is verifiable by direct substitution.
Equation 9 can be solved for ul with the boundary conditions that ul vanish at the
walls and either be periodic in x 2 and x 3 or tend to zero as x 2 and x 3 become infinite.
In order to use (8)
and (9)
even when V 2u
does not exist, we expand u)
in a
Fourier series or Fourier integral in x 2 and x 3 , our choice depending upon the bound-
d2 ( 0 )
ary conditions.
Provided that
2
exists, the individual Fourier components
dx 1
uk(x 1 ,t) exp(ik 2 x2 +ik3 x3 )
QPR No. 70
(10)
(VIII.
PHYSICAL ACOUSTICS)
for ul can be computed from (8) whether or not the full series or integral is twice differentiable with respect to x 2 and x 3 . The expansion is then a solution for the first of
the basic equations of perturbation,
6
provided that it converges
along with its first
derivative with respect to
x 3.
Substituting (10) in (9), we obtain the equation
+ik 3 Vx)(a2-k 2 ) uk(xl,t) = 0,
(
8
where a
ax1
and k
2
2
2
k 2 + k,
2
3
(11)
so that
22k2
2
k
(12)
(a -k ) ul(xl,t) = exp(-ik 3 Vx1 t) (a -k ) ul(x 1 ,0).
k
2
k
2
Note that in case k3 = 0, (11) and (12) become trivial, and ul (xl,t) = u 1 (x 1 ,0).
This is
precisely the case of perturbations that are homogeneous in the direction of
flow that was studied in general.
3
The material 5 discussed here thus involves the
behavior of the modes with k 3 f 0.
It can be verified by direct substitution that a particular inversion integral for (12)
k
2kx'
u (x 1 ,t) = e
P"
0
Equation 13,
each t,
e
2
-kx"
E(t)
2
k
exp(-ik 3 Vx"t) (a -k ) u (x",0) dx"dx'.
e
(13)
as written, vanishes identically for x I = 0, and it is clear that, for
1 > E > 0 can be chosen so that (13) will also vanish at xl = 1 and thus satisfy
the boundary conditions for
ul.
By means of estimates of the integrals in (13), certain total integrability conditions
2
on
2 u (xi,0) are derived which suffice for the uniform boundedness of Z jul(x 1,t) I
ax 1
k
and therefore of ul(xi,t).
dx
2
For example,
dx 3
either
ul(x ,0) dx1
<
ax I
or
(14)
2
dx2
dx3
u(Xi,0)
l
<co
c00
(x1 )
suffices.
As a counterexample to much weaker sufficient conditions than those of the character of (14),
QPR No. 70
a perturbation ul(xi,0) has been given,5 which possesses derivatives of all
is
(VIII.
PHYSICAL ACOUSTICS)
orders with respect to x 2 and x 3 , but only one continuous derivative with respect to x
l,
for which the solution ul(xi,t) is not bounded in time, but rather oscillates with linearly
increasing amplitude for large t.
Under certain conditions ul(xi,t) does itself tend to zero for large t,
follow that the flow is stable under these modes of perturbation.
term -Vu
in the equation
1
6
but it does not
The inhomogeneous
for u 3 yields a contribution to u 3 which fails to vanish for
large t even when ul does tend to zero, unless ul(Xi,0) vanishes identically.
The equations of perturbation
for u 2 and u 3 , for an arbitrary plane parallel flow
W(x 1 ), are
au.
at
- _W
ap
aui
ax3
6
6i3 dx1
udW
1
(15)
ax
(0)
0
can
One may easily verify that the solution for (15) with given initial conditions u
be formally written
u (x 1 ,x
2
,x 3 ,t)
= u )(x 1 ,x 2 ,x 3 -W(x 1 )t) -
f(x
1
,x 2 ,x 3 -W(x
1)(t-t'),t')
i = 2, 3,
dt'
(16)
where f represents the inhomogeneous terms on the right-hand side of (15).
It is
clear from (16)
that the u.1 cannot tend uniformly to zero for large t unless
In general, then, we conclude that plane inviscid
u0) vanish identically.
u0and
2
3
Couette flow is stable under no nontrivial perturbations whatsoever.
Actually, in view of the absence of apparent means of dissipation, the only surprising
element in this result should be the vanishing of ul for large t in certain cases.
effect can be "explained" in the following way:
basic flow,
as t -
This
as a result of the shearing action of the
oo every neighborhood of a given point (x 1 ,x 2 ,x
3
) will
contain ele-
ments of the initial perturbation field from a continuum of points along a tubular neighborhood extending from (x 1 ,x 2 ,-o0) to (X 1 ,x 2 ,oo) along the streamlines of the basic flow.
One might expect that the vanishing of ul(xi,t) at (x 1 ,x ,X3 ) as t -
0o
could result from
the vanishing of the average of the initial perturbation ul(Xi,0) along streamlines passing
near (x 1 ,x2,x
3
).
The result of the mathematical analysis is essentially the same as that given above.
It is clear that, according to the Riemann-Lebesgue theorem, the inner integral in (13),
and therefore uk(xl,t), will vanish for t - oo, provided that k3 # 0 and ax 1 uk(x,)
integrable in xl.
Because of the uniformity of convergence of
is
E juk(x 1 ,t) , u 1 (xi,t) itself will tend to
k
zero for large t,
QPR No. 70
provided that all of the ul(x 1 ,0)
for k
= 0 vanish identically.
The
(VIII.
PHYSICAL ACOUSTICS)
latter condition is clearly equivalent to
$0_
-co
ul(xi,0) dx
3
= 0
for almost all x
I
and x 2 )
which is the same condition that was stated in words on the basis of the physical argument.
H. L. Willke, Jr.
References
1. W. Heisenberg, Uber Stabilitt und Turbulenz von Fliissigkeitsstr6men,
Phys. 74, 577-627 (1924).
Ann.
2.
Lord Rayleigh, The Theory of Sound, Vol. 2 (Dover Publications, Inc., New York,
381-384.
pp.
1945),
3. H. L. Willke, Jr., Stability Problems in Parallel Flow, Ph. D. Thesis, Department of Physics, M. I. T., June 1963, pp. 17-39.
4. H. L. Willke, Jr., Stability of parallel flows, Quarterly Progress Report No. 68,
Research Laboratory of Electronics, M. I. T., January 15, 1963, p. 45.
5.
H. L.
Willke, Jr., Stability Problems in Parallel Flow, op. cit., pp. 40-77.
6.
H. L.
Willke, Jr., Stability of parallel flows,
op. cit.,
p. 45,
Eq. la.
7. It is hoped that no confusion will result from the use of k in superscripts to
stand for (k 2 ,k ).
3
B.
SOUND EMISSION FROM KARMAN VORTICES
The instability of flow past a cylinder resulting in the familiar Karman vortex street
generally represents a very weak sound source.
However, we have found that when the
cylinder is placed inside a tube and perpendicular to the axis, an intense oscillation is
produced whenever the frequency of the Karman vortex shedding coincides with one of
the cross-mode resonances in the tube.
At these frequencies the standing wave estab-
lished across a tube stimulates the instability of the flow in the tube,
and through this
feedback mechanism a self-sustained fluid oscillation of considerable magnitude results.
The frequency of shedding of the Karman vortices is known empirically to be fo = aV/d,
where V is the flow velocity and d the cylinder diameter.
The Strouhal number a is a
weak function of Reynolds number R and is approximately a constant a Z 0.18 in the
th
2< R
cross-mode
region 10 2 < R < 10 5 . In the case of a rectangular tube, the relevant n
resonance occurs at a wavelength Xn = 2D/n, where D is the transverse horizontal
dimension of the tube, which is perpendicular to the cylinder. Strong coupling between
the Karman vortex street and the transverse acoustic modes, then, occurs when f o
fn = nc/2D, where c is the speed of sound and n an integer. Using the value for fo
given above, we see that the condition for resonance can also be written
QPR No. 70
(VIII.
n~d
(V 2a
n
D
c z2.8n
d
D
PHYSICAL ACOUSTICS)
c.
Although in free space the spectrum of the sound from the Karman vortices contains
practically only the fundamental frequency f , the resonance oscillations in the tube
contain a large number of harmonics.
The feedback between the sound field and the vortex street is produced by the velocity field in the sound field.
This has been demonstrated by measuring the intensity of
the sound field as a function of position of the cylinder in the direction perpendicular to
the tube axis.
Maximum intensity is obtained when the cylinder is placed in a maxi-
mum velocity of the sound field.
A similar feedback oscillator involving two interacting cylindrical rods has also
been studied.
When two rods are placed one after the other and perpendicular to the
flow, a resonance oscillation occurs when the time of convection of a vortex from one
cylinder to the other is equal to the period of the Karman tone.
Since the drift veloc-
ity of a vortex is approximately 80 per cent of the mean flow speed in the tube, it folL
1
lows that the condition for such a resonance is simply 0 .8 V - f , which corresponds
o
to a separation of the cylinders which is equal to approximately four times the cylinder
diameter.
A more detailed account of these studies is being prepared for publication.
K. U.
C.
Ingard, W. M. Manheimer
GENERATION OF SHOCK WAVES BY MEANS OF EXPLODING
METALLIC FILMS
Shock waves produced by discharging a condenser bank through film of aluminized
Mylar have been studied in air at various pressures.
High-frequency electrostatic
transducers have been used as probes for the determination of shock speed and the
measurement of shock strength and reflection coefficients.
A preliminary analysis of
the data indicates that the shock speed is proportional to the square root of discharged
energy and, at least at low pressures, proportional to the fourth power of the ambient
density.
Also at low pressures electrical effects attributed to plasma flow behind the
shock were observed.
A detailed analysis of the data is now under way.
S. T. Emerson, A. V. Dralle, K. U. Ingard
QPR No. 70